Netto's theorem

In mathematical analysis, Netto's theorem states that continuous bijections of smooth manifolds preserve dimension.

That is, there does not exist a continuous bijection between two smooth manifolds of different dimension.

[1] The case for maps from a higher-dimensional manifold to a one-dimensional manifold was proven by Jacob Lüroth in 1878, using the intermediate value theorem to show that no manifold containing a topological circle can be mapped continuously and bijectively to the real line.

Both Netto in 1878, and Georg Cantor in 1879, gave faulty proofs of the general theorem.

[2] An important special case of this theorem concerns the non-existence of continuous bijections from one-dimensional spaces, such as the real line or unit interval, to two-dimensional spaces, such as the Euclidean plane or unit square.

The first three steps of construction of the Hilbert curve , a space-filling curve that by Netto's theorem has many self-intersections
An Osgood curve , with no self-intersections. By Netto's theorem it is impossible for such a curve to entirely cover any two-dimensional region.